A simple variational approach to weakly coupled competitive elliptic systems

Clapp, Mónica; Szulkin, Andrzej

doi:10.1007/s00030-019-0572-8

Cited by 40 publications

(40 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…, d) necessarily). This result is not new, since it re-obtains [15,Theorem 1.3] (which actually deal with possibly different powers when N ≥ 5.…”

Section: Introductionmentioning

confidence: 89%

“…In [39,40], by dividing the d components into m groups, the authors proved that system (1.1) has a least energy positive solution under appropriate assumptions on β ij . Other related and recent results in the subcritical case can be found in [14,15,16,18,29,31,47] and references, where we stress that the first two mentioned papers deal with a general p.…”

Section: Introductionmentioning

confidence: 98%

See 1 more Smart Citation

Existence of least energy positive solutions to Schrödinger systems with mixed competition and cooperation terms: the critical case

Tavares

You

2020

Calc. Var.

View full text Add to dashboard Cite

In this paper we investigate the existence of solutions to the following Schrödinger system in the critical caseHere, Ω ⊂ R 4 is a smooth bounded domain, d ≥ 2, −λ1(Ω) < λi < 0 and βii > 0 for every i, βij = βji for i = j, where λ1(Ω) is the first eigenvalue of −∆ with Dirichlet boundary conditions. Under the assumption that the components are divided into m groups, and that βij ≥ 0 (cooperation) whenever components i and j belong to the same group, while βij < 0 or βij is positive and small (competition or weak cooperation) for components i and j belonging to different groups, we establish the existence of nonnegative solutions with m nontrivial components, as well as classification results. Moreover, under additional assumptions on βij , we establish existence of least energy positive solutions in the case of mixed cooperation and competition. The proof is done by induction on the number of groups, and requires new estimates comparing energy levels of the system with those of appropriate sub-systems. In the case Ω = R 4 and λ1 = . . . = λm = 0, we present new nonexistence results. This paper can be seen as the counterpart of [Soave-Tavares, J. Differential Equations 261 (2016), 505-537] in the critical case, while extending and improving some results from [Chen-Zou, Arch.

show abstract

“…, d) necessarily). This result is not new, since it re-obtains [15,Theorem 1.3] (which actually deal with possibly different powers when N ≥ 5.…”

Section: Introductionmentioning

confidence: 89%

Section: Introductionmentioning

confidence: 98%

Existence of least energy positive solutions to Schrödinger systems with mixed competition and cooperation terms: the critical case

Tavares

You

2020

Calc. Var.

View full text Add to dashboard Cite

show abstract

“…where ω m denotes the volume of S m [3,53]. It is shown in [19,Proposition 4.6] that c = inf N J is not attained if M = R m . Therefore, from the previous paragraph, c is also not attained if M is the standard sphere S m .…”

Section: Compactness For the Yamabe Systemmentioning

confidence: 99%

Phase Separation, Optimal Partitions, and Nodal Solutions to the Yamabe Equation on the Sphere

Clapp

Saldaña

Szulkin

2020

International Mathematics Research Notices

Self Cite

View full text Add to dashboard Cite

We study an optimal $M$-partition problem for the Yamabe equation on the round sphere, in the presence of some particular symmetries. We show that there is a correspondence between solutions to this problem and least energy sign-changing symmetric solutions to the Yamabe equation on the sphere with precisely $M$ nodal domains. The existence of an optimal partition is established through the study of the limit profiles of least energy solutions to a weakly coupled competitive elliptic system on the sphere.

show abstract

“…Following the variational approach presented in the previous sections, successively Clapp and Pistoia [8], Clapp and Szulkin [12] and Clapp, Saldaña and Szulkin [11] found Γ-invariant solutions to the competitive Yamabe system (1) on S m for the groups considered by Ding, and described the limit profile of least energy solutions as λ ij → ∞. In this particular case, Theorems 1.1 and 1.2 were proved in [8,12] and [8,11] respectively. In addition, a more accurate description of the optimal (Γ, )-partition of S m is provided in [8,11].…”

mentioning

confidence: 78%

“…The variational approach introduced in [12] can be immediately adapted to establish the existence of infinitely many fully nontrivial critical points of J . We sketch this procedure.…”

mentioning

confidence: 99%

Yamabe systems and optimal partitions on manifolds with symmetries

Clapp

Pistoia

2021

era

Self Cite

View full text Add to dashboard Cite

<p style='text-indent:20px;'>We prove the existence of regular optimal <inline-formula><tex-math id="M1">\begin{document}$ G $\end{document}</tex-math></inline-formula>-invariant partitions, with an arbitrary number <inline-formula><tex-math id="M2">\begin{document}$ \ell\geq 2 $\end{document}</tex-math></inline-formula> of components, for the Yamabe equation on a closed Riemannian manifold <inline-formula><tex-math id="M3">\begin{document}$ (M,g) $\end{document}</tex-math></inline-formula> when <inline-formula><tex-math id="M4">\begin{document}$ G $\end{document}</tex-math></inline-formula> is a compact group of isometries of <inline-formula><tex-math id="M5">\begin{document}$ M $\end{document}</tex-math></inline-formula> with infinite orbits. To this aim, we study a weakly coupled competitive elliptic system of <inline-formula><tex-math id="M6">\begin{document}$ \ell $\end{document}</tex-math></inline-formula> equations, related to the Yamabe equation. We show that this system has a least energy <inline-formula><tex-math id="M7">\begin{document}$ G $\end{document}</tex-math></inline-formula>-invariant solution with nontrivial components and we show that the limit profiles of its components separate spatially as the competition parameter goes to <inline-formula><tex-math id="M8">\begin{document}$ -\infty $\end{document}</tex-math></inline-formula>, giving rise to an optimal partition. For <inline-formula><tex-math id="M9">\begin{document}$ \ell = 2 $\end{document}</tex-math></inline-formula> the optimal partition obtained yields a least energy sign-changing <inline-formula><tex-math id="M10">\begin{document}$ G $\end{document}</tex-math></inline-formula>-invariant solution to the Yamabe equation with precisely two nodal domains.</p>

show abstract

A simple variational approach to weakly coupled competitive elliptic systems

Cited by 40 publications

References 19 publications

Existence of least energy positive solutions to Schrödinger systems with mixed competition and cooperation terms: the critical case

Existence of least energy positive solutions to Schrödinger systems with mixed competition and cooperation terms: the critical case

Phase Separation, Optimal Partitions, and Nodal Solutions to the Yamabe Equation on the Sphere

Yamabe systems and optimal partitions on manifolds with symmetries

Contact Info

Product

Resources

About